A steel tape $1 \;m$ long is correctly calibrated for a temperature of $27.0\,^{\circ} C .$ The length of a steel rod measured by this tape is found to be $63.0 \;cm$ on a hot day when the temperature is $45.0\,^{\circ} C .$ What is the actual length of the steel rod on that day ? What is the length of the same steel rod on a day when the temperature is $27.0\,^oC$? Coefficient of linear expansion of steel $=1.20 \times 10^{-5}\; K ^{-1}$
A steel tape $1 \;m$ long is correctly calibrated for a temperature of $27.0\,^{\circ} C .$ The length of a steel rod measured by this tape is found to be $63.0 \;cm$ on a hot day when the temperature is $45.0\,^{\circ} C .$ What is the actual length of the steel rod on that day ? What is the length of the same steel rod on a day when the temperature is $27.0\,^oC$? Coefficient of linear expansion of steel $=1.20 \times 10^{-5}\; K ^{-1}$
Length of the steel tape at temperature $T=27^{\circ} C , l=1 m =100 cm$
At temperature $T_{1}=45^{\circ} C ,$ the length of the steel $\operatorname{rod}, l_{1}=63 cm$
Coefficient of linear expansion of steel, $\alpha=1.20 \times 10^{-5} K ^{-1}$
Let $l_{2}$ be the actual length of the steel rod and $I$ be the length of the steel tape at $45^{\circ} C .$
$l^{\prime}=l+\alpha l\left(T_{1}-T\right)$
$\therefore l^{\prime}=100+1.20 \times 10^{-5} \times 100(45-27)$
$=100.0216 cm$
Hence, the actual length of the steel rod measured by the steel tape at $45^{\circ} C$ can be calculated as
$l_{2}=\frac{100.0216}{100} \times 63=63.0136 cm$
Therefore, the actual length of the rod at $45.0\,^{\circ} C$ is $63.0136 \,cm .$ Its length at $27.0\,^{\circ} C$ is $63.0 \,cm$
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